Can my nasal epistaxis nose bleeds be resolved by surgery?

Sounds like a rather heavy source of bleeding. There are many causes of nose bleeds - usually there is a small vein/vessell sitting superficial  (on-the-surface) of the lining in your nose. Intermittently they open up and bleed. If you're on any blood thinners you may bleed easier and more heavily than average. Its easiest to see the exact source when you are actually bleeding or shortly after as the source may submerge or even disappear - especially if packs have been placed repeatedly.

Your key here is getting diagnosed - your specialist will look inside your nose with a small camera after spraying some topical anesthesia. If the source is obvious and accessible, some cautery (usally a chemical which burns the vessel) may be done in the office. If the source is far back or otherwise then an operation to burn or clip the source may be indicated. Nonetheless your problem is certainly treatable.

In the meanwhile the use of saline sprays and lubricant ointments to keep the lining of the nose moist is important particularly in winter to prevent the lining from drying out and bleeding.